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Wang et al. J. Mater. Inf. 2026, 6, 1 Page 3 of 9

Figure 1. Geometric T f across representative structural families. Different materials systems are distinguished by diverse colors, and the
evaluation criteria of T f and space groups corresponding to the structures are also provided.

ions, assuming hard-sphere ionic radii and idealized coordination geometries. Starting from a prototype
structure, the lattice is modeled as a network of touching spheres. The T f is then expressed as a dimensionless
ratio that compares the ideal cation-anion bond lengths based on ionic radii. For general structures, T f can be
reformulated using average coordination numbers or geometry-specific constants. The derivation assumes
rigid ion sizes, neglects distortions or covalency [22,23] . Therefore, the collected T f expressions vary in
mathematical form, degree of complexity, and physical interpretability.

T f serves as a simple yet powerful tool for pre-screening structural formability, guiding chemical substitution
strategies. However, their effectiveness is often limited by the rigid assumptions underlying their derivation,
such as fixed coordination environments, static ionic radii, and neglect of dynamic effects. These
oversimplifications can lead to systematic biases. For instance, the exclusion of metastable phases that are
kinetically accessible or thermodynamically competitive under synthesis conditions, or false negatives where
functionally promising materials are prematurely discarded [24,25] .